Problem: The geometric sequence $(a_i)$ is defined by the formula: $a_1 = \dfrac{16}{9}$ $a_i = \dfrac{3}{2}a_{i-1}$ What is $a_{5}$, the fifth term in the sequence?
Solution: From the given formula, we can see that the first term of the sequence is $\dfrac{16}{9}$ and the common ratio is $\dfrac{3}{2}$ To find the fifth term, we can rewrite the given recurrence as an explicit formula. The general form for a geometric sequence is $a_i = a_1 r^{i - 1}$ . In this case, we have $a_i = \dfrac{16}{9} \left(\dfrac{3}{2}\right)^{i - 1}$ To find $a_{5}$ , we can simply substitute $i = 5$ into the formula. Therefore, the fifth term is equal to $a_{5} = \dfrac{16}{9} \left(\dfrac{3}{2}\right)^{5 - 1} = 9$.